− s prim ( X, Q ) with rational coeﬃcients.\n\nH d − s ( P d Σ , C ) and H d − s ( X, C ) have pure Hodge structures, and the morphism i ∗ is com- patible with them, so that H d − s prim ( X ) gets a pure Hodge structure.\n\nThe next Proposition is the Cayley proposition.\n\nProposition 4.3. [Proposition 2.3 in [3] ] Let X = X 1 ∩⋅ ⋅ ⋅∩ X s be a quasi-smooth intersec- tion subvariety in P d Σ cut oﬀ by homogeneous polynomials f 1 . . . f s . Then for p ≠ d + s − 1 2 , d + s − 3 2\n\nRemark 4.5 . The above isomorphisms are also true with rational coeﬃcients since H ● ( X, C ) = H ● ( X, Q ) ⊗ Q C . See the beginning of Section 7.1 in